crystallographic groups

One of the first scientific applications of group theory was
to the study of the structure of crystals. In fact the work
on groups (simultaneously by Fëderov and Schoenflies in 1890)
made predictions about
crystals that were not confirmed
experimentally until much later by von Laue (1912) using X-rays.

The basic result is that the symmetry group of an infinite three-dimensional
repeating pattern is one of 230 distinct groups. The point being that the
pattern is determined by its symmetry group. Thus this solves a basic
problem in crystallography, what crystal lattices can occur.

The same problem in dimension two amounts to classifying the different types
of wallpaper patterns. This time it turns out that there are exactly
seventeen types.

One of the key ingredients in proving these results.
is the Crystallographic restriction.

Theorem A rotation of an
infinite repeating two or three-dimensional pattern has order
1,2,3,4 or 6.

This is important because it narrows the range of possibilities for
the point group of the crystal or wallpaper pattern. The point group
of a symmetry group is is the quotient group by the normal subgroup
of all translations it contains.

Very roughly speaking if you think of a crystal being composed of
atoms at points on a lattice in 3-space then the point group gives
the internal atomic symmetry, whereas the symmetry group tells
you about the entire crystal.